Saturday, 27 September 2014

[Someone asked...] The difference between Approximation and Estimation...

Estimation involves calculation, whereas Approximation is about rounding off a number.
  • The rounding off of a number can be carried out before or after the calculation.

Consider the following scenario:
The length of the door is 1.95 m and its height is 2.06 m (assuming that the dimensions given are exact).

(a) Find the area of the door and round the answer off to 2 significant figures.
The actual area of the door is 1.95 m x 2.06 m = 4.017 m^2.
Answer = 4.0 m^2 (to 2 sig fig) [Ans]

The step to round off a number (to present the final answer) is an approximation.
There is no mention of estimation in question (a), so we use the given values to compute the area first, then round off as required.

(b) Estimate the area of the door to 2 significant figures.

The purpose of doing 'estimate' or 'estimation' is to enable us carry out calculation quickly and when we don't need an exact value. A rough figure should give us a sense of the actual value. It should not be too far off from the actual value. 

To estimate, we will round the numbers off to the required "degree of accuracy" (in this case, 2 significant figures) first before doing any calculation. In other words, we are using the approximated numbers to do calculation.

Area of the door ≈ 2.0 m x 2.0 m <<<<< This is the 'estimation' step when the approximated values are used for calculation
= 4.0 m^2 [Ans]

Since 1.95 m ≈ 2.0 m (2 sig fig) and  2.06 m ≈ 2.0 (2 sig fig)

When doing Estimation, we round off the values based on the degree of accuracy given.

However, there is an exception when dealing with roots because by rounding it off to 1 SF, 2 SF, etc. may not land itself nicely to be rooted.

To estimate square root of 26.77, it does not help us to figure out what number it's 'near' to if we round it off to 1 SF (i.e. 30) or 2 SF (i.e. 27). 
However, we know that 25 (which is a perfect square) is quite close to it; so, we will estimate it to be square root of 25, and get the answer "5". 
If you key square root 26.77 in the calculator, your answer should be quite close to 5.

To extend/ combine the above.... 
If the question asks: Estimate 150.5677 x "square root of 8.343" to 2 sig fig
Then we'll have 150.5677 x "square root of 8.343" ≈ 150 x "square root 9" 
= 150 x 3
= 450 [Ans]

150.5677 when rounded off to 2 sig fig = 150
square root of 8.343 will be square root of 9 (which is the closest perfect square).

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